Quantum causal histories

TL;DR

Quantum causal histories attach finite-dimensional Hilbert spaces to events in a causal set and use local unitary mappings between spacelike-separated subsets to evolve states, forming a functor from the acausal-set poset ${\bf A}$ to Hilbert spaces. Imposing reflexivity, antisymmetry, and transitivity on the causal order translates into unitary consistency conditions that lead to directed coarse-graining invariance and a topology-like notion akin to directed topological quantum field theory. The framework provides a quantum cosmology without a single wavefunction of the universe, with density-matrix propagation constrained to complete pairs and generalisations explored via edge-based Hilbert spaces and sum-over-histories. Two concrete examples illustrate the construction: a discrete Newtonian evolution and a planar trivalent graph with edge Hilbert spaces, highlighting how local unitaries and coarse-graining shape dynamics and observables. The work lays groundwork for further interpretation and generalisation, including directed triangulation invariance and potential connections to quantum gravity models.

Abstract

Quantum causal histories are defined to be causal sets with Hilbert spaces attached to each event and local unitary evolution operators. The reflexivity, antisymmetry, and transitivity properties of a causal set are preserved in the quantum history as conditions on the evolution operators. A quantum causal history in which transitivity holds can be treated as ``directed'' topological quantum field theory. Two examples of such histories are described.

Figures (3)

• Figure 1: This is a very small causal history. $P(p)$ is the causal past of the event $p$ while $F(p)$ is its causal future. The acausal set $a$ is a complete past for $p$, and $b$ is a complete future.
• Figure 2: The acausal sets $a$ and $b$ are a complete pair.
• Figure 3: A causal set and its edge-set.